Below are five short easy to memorize Ramadan prayers that I picked from the book of Adhkaar by imām an-Nawawi. I made sure that I picked only authentic narrations (grade of Hassan and above). Below you find the Arabic, the transliteration, the translation, and some commentary when necessary.
What to say the first time you see the new crescent of Ramadan
اللَّهمَّ أَهلَّهُ علينَا بالأمنِ والإيمانِ والسَّلامةِ والإسلامِ ربِّي وربُّكَ اللَّهُ
Transliteration: Allahumma ahillahu alayna bil-amni wal-īmān was-salaamati wal-islam. Rabbi wa rabbuka Allāh
Translation: Oh Allāh, make it a start full of peace and faith, safety and Islam. My lord and your lord is Allāh
Commentary: Notice how this supplication paired peace and faith as if they are twins. In fact, in Arabic they sound alike (Amn and Emaan) because one is derived from the other. It also paired peace and Islam (Silm and Islam) because again one is derived from the other. In short, the two most important concepts in our religion (Islam and īmān) are both derived from the words that mean peace in Arabic, Amn and Silm. We ask Allāh to make this Ramadan full of peace.
What to say when breaking fast at sunset every day
ذهب الظمأ وابتلت العروق وثبت الأجر إن شاء الله
Transliteration: Dhahaba adh-Dhama' wabtallatil-urooq wa thabatal-ajr inshaa'Allāh
Translation: Thirst is gone, the veins are wet, and the reward is confirmed by the will of God
Commentary: Remember, the time of breaking the fast is one of the times where du‘ā’ is likely to be answered. So you can accompany this Prophetic du‘ā’ with any other du‘ā’ of your making.
What to say if someone tries to revile you while you're fasting
إني صائمٌ ، إني صائمٌ
Transliteration: Innee saa'im, innee saa'im
Translation: I'm fasting, I'm fasting
Commentary: imām Nawawi mentions that this either should be said loud so the aggressor can hear and hopefully stop, or it should be said in silence as a self-reminder. But the first opinion is stronger.
What to say after breaking fast with a group of people
أفطر عندكم الصائمون ، وأكل طعامكم الأبرار ، وصلت عليكم الملائكة
Transliteration: aftara indakum as-saa'imoon, wa akala ta'aamakum al-abraar, wa sallat alaikum al-malaa'ikah
Translation: May the fasting people break fast at your place, and may the pious eat from your food, and may the angels pray for you
What to say if you encounter the Night of al-Qadr
اللَّهمَّ إنَّك عفُوٌّ تُحبُّ العفوَ فاعْفُ عنِّي
Transliteration: Allahumma innaka afuwwun tuhibbu al-afwa fa'fu anni
Translation: Oh Allāh you are forgiving, and you love forgiveness, so forgive me
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